Layerwise Computability and Image Randomness

Abstract

Algorithmic randomness theory starts with a notion of an individual random object. To be reasonable, this notion should have some natural properties; in particular, an object should be random with respect to the image distribution F(P) (for some distribution P and some mapping F) if and only if it has a P-random F-preimage. This result (for computable distributions and mappings, and Martin-Löf randomness) was known for a long time (folklore); for layerwise computable mappings it was mentioned in Hoyrup and Rojas (2009, Proposition 5) (even for more general case of computable metric spaces). In this paper we provide a proof and discuss the related quantitative results and applications.

Notes

Acknowledgments

We are grateful to our colleagues in ESCAPE group (LIRMM, Montpellier), Kolmogorov seminar (Moscow), and the participants of Singapore and Heidelberg workshops, especially Jason Rute, who allowed us to include his result in the paper, for interesting discussions. We also thank the anonymous referees for their comments, suggestions, and meticulous corrections.

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